AP Calculus ABmediummcq1 pt
The function f(x) = x³ - 6x² + 9x + 1 has a local maximum at which point?
A.x = 1
B.x = 3
C.x = 0
D.x = 2
Explanation
Core Concept
First, find the derivative: f'(x) = 3x² - 12x + 9. Set it to zero: 3x² - 12x + 9 = 0, which simplifies to x² - 4x + 3 = 0. Factoring gives (x - 1)(x - 3) = 0, so critical points are at x = 1 and x = 3. Using the second derivative test: f''(x) = 6x - 12. f''(1) = -6 < 0, so x = 1 is a local maximum. f''(3) = 6 > 0, so x = 3 is a local minimum.
Correct Answer
Ax = 1
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